compiler/stdc/math/k_log.h

   1
   2 /* @(#)e_log.c 1.3 95/01/18 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  */
  13
  14 #ifndef lint
  15 static char rcsid[] = "$FreeBSD: src/lib/msun/src/k_log.h,v 1.2 2011/10/15 05:23:28 das Exp $");
  16 #endif
  17
  18 /*
  19  * k_log1p(f):
  20  * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
  21  *
  22  * The following describes the overall strategy for computing
  23  * logarithms in base e.  The argument reduction and adding the final
  24  * term of the polynomial are done by the caller for increased accuracy
  25  * when different bases are used.
  26  *
  27  * Method :
  28  *   1. Argument Reduction: find k and f such that
  29  *                      x = 2^k * (1+f),
  30  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  31  *
  32  *   2. Approximation of log(1+f).
  33  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  34  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  35  *               = 2s + s*R
  36  *      We use a special Reme algorithm on [0,0.1716] to generate
  37  *      a polynomial of degree 14 to approximate R The maximum error
  38  *      of this polynomial approximation is bounded by 2**-58.45. In
  39  *      other words,
  40  *                      2      4      6      8      10      12      14
  41  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  42  *      (the values of Lg1 to Lg7 are listed in the program)
  43  *      and
  44  *          |      2          14          |     -58.45
  45  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  46  *          |                             |
  47  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  48  *      In order to guarantee error in log below 1ulp, we compute log
  49  *      by
  50  *              log(1+f) = f - s*(f - R)        (if f is not too large)
  51  *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  52  *
  53  *      3. Finally,  log(x) = k*ln2 + log(1+f).
  54  *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  55  *         Here ln2 is split into two floating point number:
  56  *                      ln2_hi + ln2_lo,
  57  *         where n*ln2_hi is always exact for |n| < 2000.
  58  *
  59  * Special cases:
  60  *      log(x) is NaN with signal if x < 0 (including -INF) ;
  61  *      log(+INF) is +INF; log(0) is -INF with signal;
  62  *      log(NaN) is that NaN with no signal.
  63  *
  64  * Accuracy:
  65  *      according to an error analysis, the error is always less than
  66  *      1 ulp (unit in the last place).
  67  *
  68  * Constants:
  69  * The hexadecimal values are the intended ones for the following
  70  * constants. The decimal values may be used, provided that the
  71  * compiler will convert from decimal to binary accurately enough
  72  * to produce the hexadecimal values shown.
  73  */
  74
  75 static const double
  76 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  77 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  78 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  79 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  80 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  81 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  82 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  83
  84 /*
  85  * We always inline k_log1p(), since doing so produces a
  86  * substantial performance improvement (~40% on amd64).
  87  */
  88 static inline double
  89 k_log1p(double f)
  90 {
  91         double hfsq,s,z,R,w,t1,t2;
  92
  93         s = f/(2.0+f);
  94         z = s*s;
  95         w = z*z;
  96         t1= w*(Lg2+w*(Lg4+w*Lg6));
  97         t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
  98         R = t2+t1;
  99         hfsq=0.5*f*f;
 100         return s*(hfsq+R);
 101 }